Model Description (LaTeX)

Model description

The model is a sum of a pseudo-Voigt elastic peak (free mixing and FWHM), a Gaussian breathing phonon (free FWHM), a resolution-convolved DHO magnon, and a linear background:

$I(E) = I_{\text{elastic}}(E) + I_{\text{breathing}}(E) + I_{\text{magnon}}(E) + I_{\text{background}}(E)$

Individual Components

1. Elastic Peak (Pseudo-Voigt, free mixing and FWHM)

$I_{\text{elastic}}(E) = A_1 \left[ \mu \cdot L(E, x_1, \text{FWHM}_1) + (1-\mu) \cdot G(E, x_1, \text{FWHM}_1) \right]$ where $\mu$ is the mixing parameter (free), $\text{FWHM}_1$ is free, and: - $L(E, x_1, \text{FWHM}_1) = \frac{(\text{FWHM}_1/2)^2}{(E - x_1)^2 + (\text{FWHM}_1/2)^2}$ (Lorentzian) - $G(E, x_1, \text{FWHM}_1) = \exp\left(-\frac{(E - x_1)^2}{2\sigma_1^2}\right)$ where $\sigma_1 = \frac{\text{FWHM}_1}{2\sqrt{2\ln(2)}}$ (Gaussian)

2. Breathing Mode (Gaussian, free FWHM)

$I_{\text{breathing}}(E) = A_2 \exp\left(-\frac{(E - x_{2})^2}{2\sigma_2^2}\right)$ where $\sigma_2 = \frac{\text{FWHM}_2}{2\sqrt{2\ln(2)}}$ and $\text{FWHM}_2$ is free

3. Magnon Peak (DHO convolved with fixed Gaussian resolution)

$I_{\text{magnon}}(E) = \left[ \frac{A_3 (\gamma E)^2}{(E^2 - x_3^2)^2 + (E \gamma)^2} \right] \otimes G_{\text{res}}(E)$ where $\gamma = \frac{FWHM_3}{2}$ (free parameter), and $G_{\text{res}}(E)$ is a Gaussian with FWHM = 0.044 eV (fixed resolution)

4. Linear Background

$I_{\text{background}}(E) = \begin{cases} |\text{slope}| \cdot E & \text{if } E \geq 0 \\ 0 & \text{if } E < 0 \end{cases}$

Fixed Parameters

Magnon convolution resolution: FWHM = 0.044 eV

Free Parameters

Elastic peak position: $x_1$
Elastic peak amplitude: $A_1$
Elastic peak FWHM: $\text{FWHM}_1$
Elastic mixing parameter: $\mu$
Breathing peak position: $x_2$
Breathing peak amplitude: $A_2$
Breathing peak FWHM: $\text{FWHM}_2$
Magnon peak position: $x_3$
Magnon peak amplitude: $A_3$
Magnon FWHM: $\text{FWHM}_3$
Background slope: slope
Total: 11 free parameters

Parameters and Bounds

Parameter	Value
x0_1 (elastic center)	[-0.100, 0.100] eV
A_1 (elastic amplitude)	[0, 10]
res_1 (elastic FWHM)	[0.040, 0.053] eV
mu (elastic mixing)	[0.00, 1.00]
x0_2 (breathing center)	[0.050, 0.200] eV
A_2 (breathing amplitude)	[0, 10]
res_2 (breathing FWHM)	[0.040, 0.053] eV
x0_3 (magnon center)	[0.040, 0.550] eV
A_3 (magnon amplitude)	[0, inf]
res_3 (magnon FWHM)	[0.044, 10.000] eV
slope (background)	[-1, 1]

Figures for T = 15 K (1-phonon model)

Figures

The fit results (1-phonon model)

The phonon dispersion and FWHM (1-phonon model)

Phonon intensities (1-phonon model)

Figures for T = 100 K (1-phonon model)

Figures

The fit results (1-phonon model)

The phonon dispersion and FWHM (1-phonon model)

Phonon intensities (1-phonon model)

Figures for T = 150 K (1-phonon model)

Figures

The fit results (1-phonon model)

The phonon dispersion and FWHM (1-phonon model)

Phonon intensities (1-phonon model)

Figures for T = 200 K (1-phonon model)

Figures

The fit results (1-phonon model)

The phonon dispersion and FWHM (1-phonon model)

Phonon intensities (1-phonon model)

Temperature Dependence Comparison (1-phonon model)

This section compares the temperature dependence of the elastic intensity, breathing intensity, breathing dispersion, and breathing FWHM for LESCO 0.125 (1-phonon model).

Colors: 15K (blue), 100K (cyan), 150K (gold), 200K (red)
Circles: Breathing mode; Triangles: Elastic (divided by 1.5)